Is there an 아니메 in which one of the characters use a smartphone?

yes but its not an 아니메 yet but some guy from kagerou project does have a smartphone and his name is Shintarou and it is still an 망가 though

Tomoko from Watamote has a smartphone. Here's the ending. The smartphone isn't exactly a big plot device though, she just uses it to call her one friend. I forget if she plays games in class with it 또는 not 또는 if she uses a handheld game system for that.